A Classical Monetary Model - Money in the Utility Function

Jarek Hurnik

Department of Economics

Lecture III

So far, the only role played by money was to serve as a nume´raire, i.e.

we dealt with a cashless economy

We incorporate a role of money other than that of unit of account to generate demand for money

We assume that real balances are an argument in the utility function Be aware that there are other options such as cash-in-advance

So far, the only role played by money was to serve as a nume´raire, i.e.

we dealt with a cashless economy

We incorporate a role of money other than that of unit of account to generate demand for money

We assume that real balances are an argument in the utility function Be aware that there are other options such as cash-in-advance

So far, the only role played by money was to serve as a nume´raire, i.e.

we dealt with a cashless economy

We incorporate a role of money other than that of unit of account to generate demand for money

We assume that real balances are an argument in the utility function Be aware that there are other options such as cash-in-advance

Households’ preferences are now given by E₀

∞

X

t=0

β^tU(Ct,M_t Pt

, Nt) (1)

Period utility is increasing and concave in real balances^M_P^t

t

And the flow budget constraint takes the form

PtCt+ QtBt+ Mt = B_t−1+ M_t−1+ WtNt− T_t (2) By letting At ≡ B_t−1+ M_t−1denote total financial wealth at the

beginning of the period t, the flow budget constraint can be rewritten as PtCt+ QtA_t+1= A_t−1+ WtNt− T_t (3) Subject to a solvency constraint

T→∞lim E_t{A_t} ≥ 0 (4)

Households’ preferences are now given by E₀

∞

X

t=0

β^tU(Ct,M_t Pt

, Nt) (1)

Period utility is increasing and concave in real balances^M_P^t

t

And the flow budget constraint takes the form

PtCt+ QtBt+ Mt = B_t−1+ M_t−1+ WtNt− T_t (2) By letting At ≡ B_t−1+ M_t−1denote total financial wealth at the

beginning of the period t, the flow budget constraint can be rewritten as PtCt+ QtA_t+1= A_t−1+ WtNt− T_t (3) Subject to a solvency constraint

T→∞lim E_t{A_t} ≥ 0 (4)

Households’ preferences are now given by E₀

∞

X

t=0

β^tU(Ct,M_t Pt

, Nt) (1)

Period utility is increasing and concave in real balances^M_P^t

t

And the flow budget constraint takes the form

PtCt+ QtBt+ Mt = B_t−1+ M_t−1+ WtNt− T_t (2) By letting At ≡ B_t−1+ M_t−1denote total financial wealth at the

beginning of the period t, the flow budget constraint can be rewritten as PtCt+ QtA_t+1= A_t−1+ WtNt− T_t (3) Subject to a solvency constraint

T→∞lim E_t{A_t} ≥ 0 (4)

Households’ preferences are now given by E₀

∞

X

t=0

β^tU(Ct,M_t Pt

, Nt) (1)

Period utility is increasing and concave in real balances^M_P^t

t

And the flow budget constraint takes the form

PtCt+ QtBt+ Mt = B_t−1+ M_t−1+ WtNt− T_t (2) By letting At ≡ B_t−1+ M_t−1denote total financial wealth at the

beginning of the period t, the flow budget constraint can be rewritten as PtCt+ QtA_t+1= A_t−1+ WtNt− T_t (3) Subject to a solvency constraint

T→∞lim E_t{A_t} ≥ 0 (4)

Using At it is assumed that all financial assets yield gross nominal return Q⁻¹(= exp{it})

Households purchase the utility-yielding ’services’ of money balances at a unit price (1 − Qt) = 1 − exp{−it} ' i_t

The implicit price of money services roughly corresponds to the nominal interest rate, which in turn is the opportunity cost of holding money

Using At it is assumed that all financial assets yield gross nominal return Q⁻¹(= exp{it})

Households purchase the utility-yielding ’services’ of money balances at a unit price (1 − Qt) = 1 − exp{−it} ' i_t

The implicit price of money services roughly corresponds to the nominal interest rate, which in turn is the opportunity cost of holding money

Using At it is assumed that all financial assets yield gross nominal return Q⁻¹(= exp{it})

Households purchase the utility-yielding ’services’ of money balances at a unit price (1 − Qt) = 1 − exp{−it} ' i_t

The implicit price of money services roughly corresponds to the nominal interest rate, which in turn is the opportunity cost of holding money

Two of the optimality conditions are same as those for the cashless model

−U_n,t

U_c,t = W_t Pt

(5) Q_t = βEt

 U_c,t+1 U_c,t

P_t P_t+1



(6) And there is an additional optimality condition given by

U_m,t

U_c,t = 1 − exp{−it} (7)

For any statement about consequences of having money in the utility function , more precision is needed about the way money balances interact with other variables in yielding utility

Two of the optimality conditions are same as those for the cashless model

−U_n,t

U_c,t = W_t Pt

(5) Q_t = βEt

 U_c,t+1 U_c,t

P_t P_t+1



(6) And there is an additional optimality condition given by

U_m,t

U_c,t = 1 − exp{−it} (7)

For any statement about consequences of having money in the utility function , more precision is needed about the way money balances interact with other variables in yielding utility

Two of the optimality conditions are same as those for the cashless model

−U_n,t

U_c,t = W_t Pt

(5) Q_t = βEt

 U_c,t+1 U_c,t

P_t P_t+1



(6) And there is an additional optimality condition given by

U_m,t

U_c,t = 1 − exp{−it} (7)

For any statement about consequences of having money in the utility function , more precision is needed about the way money balances interact with other variables in yielding utility

Assume following functional form U(Ct,Mt

P_t, Nt) = C^1−σ_t

1 − σ+(Mt/Pt)^1−ν

1 − ν − N_t^1+ϕ

1 + ϕ (8)

Given assumed separability neither U_c,tnor U_n,tdepend on the level of real balances

As a result

C_t^σN_t^ϕ = Wt

Pt

(9) Q_t = βEt

( C_t+1 C_t

−σ

P_t P_t+1

)

(10) remain unchanged and so do their log-linear counterparts

c_t = E_t{c_t+1} − 1

σ (it− E_t{π_t+1}) (11)

wt− p_t = σct+ ϕnt (12)

Assume following functional form U(Ct,Mt

P_t, Nt) = C^1−σ_t

1 − σ+(Mt/Pt)^1−ν

1 − ν − N_t^1+ϕ

1 + ϕ (8)

Given assumed separability neither U_c,tnor U_n,tdepend on the level of real balances

As a result

C_t^σN_t^ϕ = Wt

Pt

(9) Q_t = βEt

( C_t+1 C_t

−σ

P_t P_t+1

)

(10) remain unchanged and so do their log-linear counterparts

c_t = E_t{c_t+1} − 1

σ (it− E_t{π_t+1}) (11)

wt− p_t = σct+ ϕnt (12)

Assume following functional form U(Ct,Mt

P_t, Nt) = C^1−σ_t

1 − σ+(Mt/Pt)^1−ν

1 − ν − N_t^1+ϕ

1 + ϕ (8)

Given assumed separability neither U_c,tnor U_n,tdepend on the level of real balances

As a result

C_t^σN_t^ϕ = Wt

Pt

(9) Q_t = βEt

( C_t+1 C_t

−σ

P_t P_t+1

)

(10) remain unchanged and so do their log-linear counterparts

c_t = E_t{c_t+1} − 1

σ (it− E_t{π_t+1}) (11)

wt− p_t = σct+ ϕnt (12)

Note that equilibrium values for output, employment, the real rate, and the real wage are determined in the same way as in cashless economy However, introduction of money in utility function allows a money demand equation to be derived from the households’ optimal behaviour Given the specification of utility, the new optimality condition can be rewritten as

Mt

Pt

= C_t^σ/ν

1 − exp{−it}^−1/ν

(13) And in approximate log-linear form as

m_t− p_t = σ

νc_t− ηi_t (14)

where η ≡ ν(exp{¯i}−1)¹ ' _ν¯i¹

Note that equilibrium values for output, employment, the real rate, and the real wage are determined in the same way as in cashless economy However, introduction of money in utility function allows a money demand equation to be derived from the households’ optimal behaviour Given the specification of utility, the new optimality condition can be rewritten as

Mt

Pt

= C_t^σ/ν

1 − exp{−it}^−1/ν

(13) And in approximate log-linear form as

m_t− p_t = σ

νc_t− ηi_t (14)

where η ≡ ν(exp{¯i}−1)¹ ' _ν¯i¹

Note that equilibrium values for output, employment, the real rate, and the real wage are determined in the same way as in cashless economy However, introduction of money in utility function allows a money demand equation to be derived from the households’ optimal behaviour Given the specification of utility, the new optimality condition can be rewritten as

Mt

Pt

= C_t^σ/ν

1 − exp{−it}^−1/ν

(13) And in approximate log-linear form as

m_t− p_t = σ

νc_t− ηi_t (14)

where η ≡ ν(exp{¯i}−1)¹ ' _ν¯i¹

Note that equilibrium values for output, employment, the real rate, and the real wage are determined in the same way as in cashless economy However, introduction of money in utility function allows a money demand equation to be derived from the households’ optimal behaviour Given the specification of utility, the new optimality condition can be rewritten as

Mt

Pt

= C_t^σ/ν

1 − exp{−it}^−1/ν

(13) And in approximate log-linear form as

m_t− p_t = σ

νc_t− ηi_t (14)

where η ≡ ν(exp{¯i}−1)¹ ' _ν¯i¹

The particular case of ν = σ is an appealing one, because it implies a unit elasticity with respect to consumption

This assumption yields a conventional linear demand for money

mt− p_t = ct− ηi_t (15)

= y_t− ηi_t (16)

assuming that all output is consumed

(15) determines the equilibrium values for inflation and other nominal variables whenever the description of monetary policy involves the quantity of money

Otherwise (15) determines the quantity of money that the central bank will need to supply in order to support the nominal interest rate implied by the policy rule

The particular case of ν = σ is an appealing one, because it implies a unit elasticity with respect to consumption

This assumption yields a conventional linear demand for money

mt− p_t = ct− ηi_t (15)

= y_t− ηi_t (16)

assuming that all output is consumed

(15) determines the equilibrium values for inflation and other nominal variables whenever the description of monetary policy involves the quantity of money

Otherwise (15) determines the quantity of money that the central bank will need to supply in order to support the nominal interest rate implied by the policy rule

The particular case of ν = σ is an appealing one, because it implies a unit elasticity with respect to consumption

This assumption yields a conventional linear demand for money

mt− p_t = ct− ηi_t (15)

= y_t− ηi_t (16)

assuming that all output is consumed

(15) determines the equilibrium values for inflation and other nominal variables whenever the description of monetary policy involves the quantity of money

Otherwise (15) determines the quantity of money that the central bank will need to supply in order to support the nominal interest rate implied by the policy rule

The particular case of ν = σ is an appealing one, because it implies a unit elasticity with respect to consumption

This assumption yields a conventional linear demand for money

mt− p_t = ct− ηi_t (15)

= y_t− ηi_t (16)

assuming that all output is consumed

(15) determines the equilibrium values for inflation and other nominal variables whenever the description of monetary policy involves the quantity of money

Otherwise (15) determines the quantity of money that the central bank will need to supply in order to support the nominal interest rate implied by the policy rule

Note that equilibrium values for output, employment, the real rate, and the real wage are determined in the same way as in cashless economy However, introduction of money in utility function allows a money demand equation to be derived from the households’ optimal behaviour Given the specification of utility, the new optimality condition can be rewritten as

Mt

Pt

= C_t^σ/ν

1 − exp{−it}^−1/ν

(17) And in approximate log-linear form as

m_t− p_t = σ

νc_t− ηi_t (18)

where η ≡ ν(exp{¯i}−1)¹ ' _ν¯i¹

Note that equilibrium values for output, employment, the real rate, and the real wage are determined in the same way as in cashless economy However, introduction of money in utility function allows a money demand equation to be derived from the households’ optimal behaviour Given the specification of utility, the new optimality condition can be rewritten as

Mt

Pt

= C_t^σ/ν

1 − exp{−it}^−1/ν

(17) And in approximate log-linear form as

m_t− p_t = σ

νc_t− ηi_t (18)

where η ≡ ν(exp{¯i}−1)¹ ' _ν¯i¹

Note that equilibrium values for output, employment, the real rate, and the real wage are determined in the same way as in cashless economy However, introduction of money in utility function allows a money demand equation to be derived from the households’ optimal behaviour Given the specification of utility, the new optimality condition can be rewritten as

Mt

Pt

= C_t^σ/ν

1 − exp{−it}^−1/ν

(17) And in approximate log-linear form as

m_t− p_t = σ

νc_t− ηi_t (18)

where η ≡ ν(exp{¯i}−1)¹ ' _ν¯i¹

Note that equilibrium values for output, employment, the real rate, and the real wage are determined in the same way as in cashless economy However, introduction of money in utility function allows a money demand equation to be derived from the households’ optimal behaviour Given the specification of utility, the new optimality condition can be rewritten as

Mt

Pt

= C_t^σ/ν

1 − exp{−it}^−1/ν

(17) And in approximate log-linear form as

m_t− p_t = σ

νc_t− ηi_t (18)

where η ≡ ν(exp{¯i}−1)¹ ' _ν¯i¹

Let now consider an economy in which period utility is given U(Ct,M_t

Pt

, Nt) = X^1−σ_t

1 − σ− N_t^1+ϕ

1 + ϕ (19)

where Xt is a composite index of consumption and real balances

Xt ≡

"

(1 − ϑ) C^1−ν_t + ϑ Mt

Pt

1−ν#_1−ν¹

≡ C_t^1−ϑ Mt

Pt

ϑ

for ν = 1

ν represents the (inverse) elasticity of substitution between consumption and real balances, and ϑ the relative weight of real balances in utility

Let now consider an economy in which period utility is given U(Ct,M_t

Pt

, Nt) = X^1−σ_t

1 − σ− N_t^1+ϕ

1 + ϕ (19)

where Xt is a composite index of consumption and real balances

Xt ≡

"

(1 − ϑ) C^1−ν_t + ϑ Mt

Pt

1−ν#_1−ν¹

≡ C_t^1−ϑ Mt

Pt

ϑ

for ν = 1

ν represents the (inverse) elasticity of substitution between consumption and real balances, and ϑ the relative weight of real balances in utility

Let now consider an economy in which period utility is given U(Ct,M_t

Pt

, Nt) = X^1−σ_t

1 − σ− N_t^1+ϕ

1 + ϕ (19)

where Xt is a composite index of consumption and real balances

Xt ≡

"

(1 − ϑ) C^1−ν_t + ϑ Mt

Pt

1−ν#_1−ν¹

≡ C_t^1−ϑ Mt

Pt

ϑ

for ν = 1

ν represents the (inverse) elasticity of substitution between consumption and real balances, and ϑ the relative weight of real balances in utility

Marginal utilities of consumption and real balances are now given by U_c,t = (1 − ϑ) X_t^ν−σC^−ν_t

U_m,t = ϑX_t^ν−σ Mt

Pt

−ϑ

whereas marginal utility of labour is, as before, given by U_n,t = −N_t^ϕ Optimality conditions of the household’s problem are now

Wt

Pt

= N_t^ϕX_t^ν−σC^ν_t(1 − ϑ)⁻¹ (20) Qt = βEt

( C_t+1 C_t

−ν

 X_t+1 X_t

ν−σ

Pt

P_t+1 )

(21) M_t

Pt

= C_t(1 − exp{−it})^−1/ν

 ϑ

1 − ϑ

_ν¹

(22)

Marginal utilities of consumption and real balances are now given by U_c,t = (1 − ϑ) X_t^ν−σC^−ν_t

U_m,t = ϑX_t^ν−σ Mt

Pt

−ϑ

whereas marginal utility of labour is, as before, given by U_n,t = −N_t^ϕ Optimality conditions of the household’s problem are now

Wt

Pt

= N_t^ϕX_t^ν−σC^ν_t(1 − ϑ)⁻¹ (20) Qt = βEt

( C_t+1 C_t

−ν

 X_t+1 X_t

ν−σ

Pt

P_t+1 )

(21) M_t

Pt

= C_t(1 − exp{−it})^−1/ν

 ϑ

1 − ϑ

_ν¹

(22)

Optimality conditions imply that monetary policy is no longer neutral Real money balances influence labour supply as well as consumption Whereas depend on the nominal interest rate

Different paths of the interest rate have different implications for real balances, consumption and labour supply

Formally, we start noticing that the implied money demand equation (22) has following log-linear form

m_t− p_t = ct− ηi_t (23)

where η ≡ ν(exp{¯i}−1)¹

Money demand semi-elasticity depends on the elasticity of substitution between real balances and consumption ν⁻¹

Optimality conditions imply that monetary policy is no longer neutral Real money balances influence labour supply as well as consumption Whereas depend on the nominal interest rate

Different paths of the interest rate have different implications for real balances, consumption and labour supply

Formally, we start noticing that the implied money demand equation (22) has following log-linear form

m_t− p_t = ct− ηi_t (23)

where η ≡ ν(exp{¯i}−1)¹

Money demand semi-elasticity depends on the elasticity of substitution between real balances and consumption ν⁻¹

Optimality conditions imply that monetary policy is no longer neutral Real money balances influence labour supply as well as consumption Whereas depend on the nominal interest rate

Different paths of the interest rate have different implications for real balances, consumption and labour supply

Formally, we start noticing that the implied money demand equation (22) has following log-linear form

m_t− p_t = ct− ηi_t (23)

where η ≡ ν(exp{¯i}−1)¹

Money demand semi-elasticity depends on the elasticity of substitution between real balances and consumption ν⁻¹

Optimality conditions imply that monetary policy is no longer neutral Real money balances influence labour supply as well as consumption Whereas depend on the nominal interest rate

Different paths of the interest rate have different implications for real balances, consumption and labour supply

Formally, we start noticing that the implied money demand equation (22) has following log-linear form

m_t− p_t = ct− ηi_t (23)

where η ≡ ν(exp{¯i}−1)¹

Money demand semi-elasticity depends on the elasticity of substitution between real balances and consumption ν⁻¹

Log-linearization of 20 yields

wt− pt = σct+ ϕnt+ (ν − σ)(ct− xt) (24) Log-linearizing expression fo Xt, combining with 22 and substituting for xt above yields

wt− p_t = σct+ ϕnt+ χ(ν − σ)(ct− (m_t− p_t)) (25) where χ = ^ϑ

ν1(1−β)^{1− 1ν}

(1−ϑ)^ν1ϑ^ν1(1−β)^{1− 1ν}

Finally, substituting for ct− (m_t− p_t) from log-linear money demand yields

w_t− p_t = σct+ ϕnt+ χη(ν − σ)it (26)

Log-linearization of 20 yields

wt− pt = σct+ ϕnt+ (ν − σ)(ct− xt) (24) Log-linearizing expression fo Xt, combining with 22 and substituting for xt above yields

wt− p_t = σct+ ϕnt+ χ(ν − σ)(ct− (m_t− p_t)) (25) where χ = ^ϑ

ν1(1−β)^{1− 1ν}

(1−ϑ)^ν1ϑ^ν1(1−β)^{1− 1ν}

Finally, substituting for ct− (m_t− p_t) from log-linear money demand yields

w_t− p_t = σct+ ϕnt+ χη(ν − σ)it (26)

Log-linearization of 20 yields

wt− pt = σct+ ϕnt+ (ν − σ)(ct− xt) (24) Log-linearizing expression fo Xt, combining with 22 and substituting for xt above yields

wt− p_t = σct+ ϕnt+ χ(ν − σ)(ct− (m_t− p_t)) (25) where χ = ^ϑ

ν1(1−β)^{1− 1ν}

(1−ϑ)^ν1ϑ^ν1(1−β)^{1− 1ν}

Finally, substituting for ct− (m_t− p_t) from log-linear money demand yields

w_t− p_t = σct+ ϕnt+ χη(ν − σ)it (26)

Let’s define steady-state ratio of real balances to consumption km≡ ^M/¯¯_¯^P

C

Then using money demand equation one gets km=

 ϑ

(1−β)(1−ϑ)

¹

ν and χ = _1+k^km(1−β)

m(1−β)

Multiplying χ, η and ν − σ one gets ω ≡ ^kmβ(^1−σ_ν)

1+km(1−β) and

w_t− p_t = σct+ ϕnt+ ωit (27) Impact of interest rate on labour supply depends on the sign of ω and that is determined by the sign of ν − σ

When ν > σ (implying ω > 0) the reduction in real balances (increase in interest rate) brings down marginal utility of consumption, lowering the quantity of labour supplied

Let’s define steady-state ratio of real balances to consumption km≡ ^M/¯¯_¯^P

C

Then using money demand equation one gets km=

 ϑ

(1−β)(1−ϑ)

¹

ν and

χ = _1+k^km(1−β)

m(1−β)

Multiplying χ, η and ν − σ one gets ω ≡ ^kmβ(^1−σ_ν)

1+km(1−β) and

w_t− p_t = σct+ ϕnt+ ωit (27) Impact of interest rate on labour supply depends on the sign of ω and that is determined by the sign of ν − σ

When ν > σ (implying ω > 0) the reduction in real balances (increase in interest rate) brings down marginal utility of consumption, lowering the quantity of labour supplied

Let’s define steady-state ratio of real balances to consumption km≡ ^M/¯¯_¯^P

C

Then using money demand equation one gets km=

 ϑ

(1−β)(1−ϑ)

¹

ν and

χ = _1+k^km(1−β)

m(1−β)

Multiplying χ, η and ν − σ one gets ω ≡ ^kmβ(^1−σ_ν)

1+km(1−β) and

w_t− p_t = σct+ ϕnt+ ωit (27) Impact of interest rate on labour supply depends on the sign of ω and that is determined by the sign of ν − σ

When ν > σ (implying ω > 0) the reduction in real balances (increase in interest rate) brings down marginal utility of consumption, lowering the quantity of labour supplied

Let’s define steady-state ratio of real balances to consumption km≡ ^M/¯¯_¯^P

C

Then using money demand equation one gets km=

 ϑ

(1−β)(1−ϑ)

¹

ν and

χ = _1+k^km(1−β)

m(1−β)

Multiplying χ, η and ν − σ one gets ω ≡ ^kmβ(^1−σ_ν)

1+km(1−β) and

w_t− p_t = σct+ ϕnt+ ωit (27) Impact of interest rate on labour supply depends on the sign of ω and that is determined by the sign of ν − σ

When ν > σ (implying ω > 0) the reduction in real balances (increase in interest rate) brings down marginal utility of consumption, lowering the quantity of labour supplied

Let’s define steady-state ratio of real balances to consumption km≡ ^M/¯¯_¯^P

C

Then using money demand equation one gets km=

 ϑ

(1−β)(1−ϑ)

¹

ν and

χ = _1+k^km(1−β)

m(1−β)

Multiplying χ, η and ν − σ one gets ω ≡ ^kmβ(^1−σ_ν)

1+km(1−β) and

w_t− p_t = σct+ ϕnt+ ωit (27) Impact of interest rate on labour supply depends on the sign of ω and that is determined by the sign of ν − σ

When ν > σ (implying ω > 0) the reduction in real balances (increase in interest rate) brings down marginal utility of consumption, lowering the quantity of labour supplied

Log-linear approximation of the Euler equation (21) looks as follows ct = Et{c_t+1} − 1

σ(it− Et{π_t+1}

− (ν − σ)E_t{(c_t+1− x_t+1) − (ct− x_t)} − ρ) ct = Et{c_t+1} − 1

σ(it− E_t{π_t+1}

− χ(ν − σ)Et{∆c_t+1− ∆(m_t+1− p_t+1)} − ρ) c_t = E_t{c_t+1} − 1

σ (it− E_t{π_t+1} − ωE_t{∆i_t+1} − ρ) (28) Anticipation of a nominal interest rate increase lowers the expected level of the marginal utility of consumption and induces an increase in current consumption

Log-linear approximation of the Euler equation (21) looks as follows ct = Et{c_t+1} − 1

σ(it− Et{π_t+1}

− (ν − σ)E_t{(c_t+1− x_t+1) − (ct− x_t)} − ρ) ct = Et{c_t+1} − 1

σ(it− E_t{π_t+1}

− χ(ν − σ)Et{∆c_t+1− ∆(m_t+1− p_t+1)} − ρ) c_t = E_t{c_t+1} − 1

σ (it− E_t{π_t+1} − ωE_t{∆i_t+1} − ρ) (28) Anticipation of a nominal interest rate increase lowers the expected level of the marginal utility of consumption and induces an increase in current consumption

Let’s start by combining labour supply and labour demand which both equal the real wage

σct+ ϕnt+ ωit = yt− nt+ log(1 − α) (29) Then using market clearing condition yt = ctand production function yt = at+ αntyields

yt= ψyaat+ ψyiit+ υya (30) where ψyi≡ _{σ+ϕ+α(1−σ)}^{ω(1−alpha)}

Equilibrium output is no more invariant to monetary policy, money is not neutral and the above equilibrium condition does not suffice to determine output

Let’s start by combining labour supply and labour demand which both equal the real wage

σct+ ϕnt+ ωit = yt− nt+ log(1 − α) (29) Then using market clearing condition yt = ctand production function yt = at+ αntyields

yt= ψyaat+ ψyiit+ υya (30) where ψyi≡ _{σ+ϕ+α(1−σ)}^{ω(1−alpha)}

Equilibrium output is no more invariant to monetary policy, money is not neutral and the above equilibrium condition does not suffice to determine output

Let’s start by combining labour supply and labour demand which both equal the real wage

σct+ ϕnt+ ωit = yt− nt+ log(1 − α) (29) Then using market clearing condition yt = ctand production function yt = at+ αntyields

yt= ψyaat+ ψyiit+ υya (30) where ψyi≡ _{σ+ϕ+α(1−σ)}^{ω(1−alpha)}

Equilibrium output is no more invariant to monetary policy, money is not neutral and the above equilibrium condition does not suffice to determine output

In order to pin down equilibrium path of endogenous variables

equilibrium condition for output has to be combined with Euler equation and description of monetary policy

y_t = E_t{y_t+1} − 1

σ(it− E_t{π_t+1} − ωE_t{∆i_t+1} − ρ) (31)

it = ρ + Φ_ππt+ υt (32)

where we assume that υ follows a stationary AR(1) process υt = ρυυt−1+ ^υ_t

Similarly assume that the technology follows the AR(1) process at = ρaat−1+ ^a_t

In order to pin down equilibrium path of endogenous variables

equilibrium condition for output has to be combined with Euler equation and description of monetary policy

y_t = E_t{y_t+1} − 1

σ(it− E_t{π_t+1} − ωE_t{∆i_t+1} − ρ) (31)

it = ρ + Φ_ππt+ υt (32)

where we assume that υ follows a stationary AR(1) process υt = ρυυt−1+ ^υ_t

Similarly assume that the technology follows the AR(1) process at = ρaat−1+ ^a_t

Closed form expression for the equilibrium level of inflation, interest rate and output looks as follows

πt = − σ(1 − ρa)ψya

Φπ(1 + ωψ)(1 − Θρa)at− 1 + (1 − ρυ)ωψ Φπ(1 + ωψ)(1 − Θρυ)υt

it = − σ(1 − ρa)ψya

(1 + ωψ)(1 − Θρa)at− ρ_υ

Φπ(1 + ωψ)(1 − Θρυ)υt

yt = ψya



1 + σ(1 − ρa)ψyi

(1 + ωψ)(1 − Θρa)



at+ ρ_υψyi

Φπ(1 + ωψ)(1 − Θρυ)υt

where Θ ≡ _(1+ωψ)Φ^1+ωψΦπ

π and ψ ≡ _{σ(1−α)+α+ϕ}^α+ϕ

Interest rate multiplier of output, conditional on an exogenous monetary policy shock, is given by ^dy_di^t

t = ^dy_di^t/dυt

t/dυt = −ψyi

In order to get sense for the magnitude, recall that ψyi≡ _{σ+ϕ+α(1−σ)}^{ω(1−alpha)} . Assuming common calibration σ = ϕ = 1 and α = 1/3, one gets ψyi= 1/3ω

Having in mind that ω itself depends mainly on km, especially when β is close to one and η is relatively small, one can approximate ψyi= 1/3km

Size of kmdepends on the definition of money and ranges from km' 0.3 to km' 3 for monetary base and M2 respectively

Interest rate multiplier of output, conditional on an exogenous monetary policy shock, is given by ^dy_di^t

t = ^dy_di^t/dυt

t/dυt = −ψyi

In order to get sense for the magnitude, recall that ψyi≡ _{σ+ϕ+α(1−σ)}^{ω(1−alpha)} . Assuming common calibration σ = ϕ = 1 and α = 1/3, one gets ψyi= 1/3ω

Having in mind that ω itself depends mainly on km, especially when β is close to one and η is relatively small, one can approximate ψyi= 1/3km

Size of kmdepends on the definition of money and ranges from km' 0.3 to km' 3 for monetary base and M2 respectively

Interest rate multiplier of output, conditional on an exogenous monetary policy shock, is given by ^dy_di^t

t = ^dy_di^t/dυt

t/dυt = −ψyi

In order to get sense for the magnitude, recall that ψyi≡ _{σ+ϕ+α(1−σ)}^{ω(1−alpha)} . Assuming common calibration σ = ϕ = 1 and α = 1/3, one gets ψyi= 1/3ω

Having in mind that ω itself depends mainly on km, especially when β is close to one and η is relatively small, one can approximate ψyi= 1/3km

Size of kmdepends on the definition of money and ranges from km' 0.3 to km' 3 for monetary base and M2 respectively

Interest rate multiplier of output, conditional on an exogenous monetary policy shock, is given by ^dy_di^t

t = ^dy_di^t/dυt

t/dυt = −ψyi

In order to get sense for the magnitude, recall that ψyi≡ _{σ+ϕ+α(1−σ)}^{ω(1−alpha)} . Assuming common calibration σ = ϕ = 1 and α = 1/3, one gets ψyi= 1/3ω

Having in mind that ω itself depends mainly on km, especially when β is close to one and η is relatively small, one can approximate ψyi= 1/3km

Size of kmdepends on the definition of money and ranges from km' 0.3 to km' 3 for monetary base and M2 respectively

Leading to values of ψyiin the range from 0.1 to 1 and impact of one percent increase in interest rate (annualized) in the range from 0.025 to 0.25 percent.

The latter value, while small, appears to be closer to the estimated output effects of a monetary policy shock found in the literature

However, there are other aspects of the transmission of monetary shocks that are at odds with the evidence.

Note that dπt

dit

= dπt/dυt

dit/dυt

= (1 + (1 − ρ_υ)ωψρ⁻¹_υ > 0 dr_t

dit

= 1 −dE_t{π_t+1}/dυ_t dit/dυt

= −(1 − ρυ)ωψ < 0

Leading to values of ψyiin the range from 0.1 to 1 and impact of one percent increase in interest rate (annualized) in the range from 0.025 to 0.25 percent.

The latter value, while small, appears to be closer to the estimated output effects of a monetary policy shock found in the literature

However, there are other aspects of the transmission of monetary shocks that are at odds with the evidence.

Note that dπt

dit

= dπt/dυt

dit/dυt

= (1 + (1 − ρ_υ)ωψρ⁻¹_υ > 0 dr_t

dit

= 1 −dE_t{π_t+1}/dυ_t dit/dυt

= −(1 − ρυ)ωψ < 0

Leading to values of ψyiin the range from 0.1 to 1 and impact of one percent increase in interest rate (annualized) in the range from 0.025 to 0.25 percent.

The latter value, while small, appears to be closer to the estimated output effects of a monetary policy shock found in the literature

However, there are other aspects of the transmission of monetary shocks that are at odds with the evidence.

Note that dπt

dit

= dπt/dυt

dit/dυt

= (1 + (1 − ρ_υ)ωψρ⁻¹_υ > 0 dr_t

dit

= 1 −dE_t{π_t+1}/dυ_t dit/dυt

= −(1 − ρυ)ωψ < 0

Long-run output effects of monetary policy that permanently raises nominal interest rate is same as the short-run, i.e. −ψyi

Any permanent increase in inflation and nominal interest rate leads to lower output

In principle there is no problem with that, however, the lack of significant empirical relationship between long-run inflation and economic activity (at least at low levels of inflation), suggests a low value for kmand ψyi

Unfortunately, negligible long-run tradeoff is associated with negligible short run effects of monetary policy

Long-run output effects of monetary policy that permanently raises nominal interest rate is same as the short-run, i.e. −ψyi

Any permanent increase in inflation and nominal interest rate leads to lower output

In principle there is no problem with that, however, the lack of significant empirical relationship between long-run inflation and economic activity (at least at low levels of inflation), suggests a low value for kmand ψyi

Unfortunately, negligible long-run tradeoff is associated with negligible short run effects of monetary policy

Long-run output effects of monetary policy that permanently raises nominal interest rate is same as the short-run, i.e. −ψyi

Any permanent increase in inflation and nominal interest rate leads to lower output

In principle there is no problem with that, however, the lack of significant empirical relationship between long-run inflation and economic activity (at least at low levels of inflation), suggests a low value for kmand ψyi

Unfortunately, negligible long-run tradeoff is associated with negligible short run effects of monetary policy

Long-run output effects of monetary policy that permanently raises nominal interest rate is same as the short-run, i.e. −ψyi

Any permanent increase in inflation and nominal interest rate leads to lower output

In principle there is no problem with that, however, the lack of significant empirical relationship between long-run inflation and economic activity (at least at low levels of inflation), suggests a low value for kmand ψyi

Unfortunately, negligible long-run tradeoff is associated with negligible short run effects of monetary policy

Hypothetical social planner seeking to maximize the utility of

representative household would solve a sequence of static problems of the form

maxU(Ct,Mt

P_t, Nt) subject to the resource constraint

Ct = AtN_t^1−α

The optimality conditions for that problem are given

−U_n,t

U_c,t = (1 − α)AtN_t^−α (33)

U_m,t = 0 (34)

the latter condition equates marginal utility of real balances to the social marginal cost of their production, which is implicitly assumed to be zero

As household’s optimal choice of money balances requires

−U_m,t

U_c,t = (1 − exp{−it})

efficiency condition (34 will be satisfied, if and only if, it = 0 for all t, a policy known as the Friedman rule

Note that such a policy implies an average rate of inflation π = −ρ < 0

i.e., prices will decline on average at the rate of time preference

As household’s optimal choice of money balances requires

−U_m,t

U_c,t = (1 − exp{−it})

efficiency condition (34 will be satisfied, if and only if, it = 0 for all t, a policy known as the Friedman rule

Note that such a policy implies an average rate of inflation π = −ρ < 0

i.e., prices will decline on average at the rate of time preference

A policy rule of the form it= 0 for all t leads to price level

indeterminacy, so the central bank cannot just set it = 0 for all t in order to implement the Friedman rule

By following a rule of the form

it = Φ(r_t−1− π_t)

the central bank can avoid price level indeterminacy, while setting iton average equal to zero

To see this, combine the above rule with Fisher equation it= rt+ Et{π_t+1}, which implies the difference equation

E_t{i_t+1} = Φi_t (35)

whose only stationary solution is it = 0 for all t

Under the above rule inflation is fully predictable and given by

πt = −r_t−1 (36)

A policy rule of the form it= 0 for all t leads to price level

indeterminacy, so the central bank cannot just set it = 0 for all t in order to implement the Friedman rule

By following a rule of the form

it = Φ(r_t−1− π_t)

the central bank can avoid price level indeterminacy, while setting iton average equal to zero

To see this, combine the above rule with Fisher equation it= rt+ Et{π_t+1}, which implies the difference equation

E_t{i_t+1} = Φi_t (35)

whose only stationary solution is it = 0 for all t

Under the above rule inflation is fully predictable and given by

πt = −r_t−1 (36)

A policy rule of the form it= 0 for all t leads to price level

indeterminacy, so the central bank cannot just set it = 0 for all t in order to implement the Friedman rule

By following a rule of the form

it = Φ(r_t−1− π_t)

the central bank can avoid price level indeterminacy, while setting iton average equal to zero

To see this, combine the above rule with Fisher equation it= rt+ Et{π_t+1}, which implies the difference equation

E_t{i_t+1} = Φi_t (35)

whose only stationary solution is it = 0 for all t

Under the above rule inflation is fully predictable and given by

πt = −r_t−1 (36)

A policy rule of the form it= 0 for all t leads to price level

indeterminacy, so the central bank cannot just set it = 0 for all t in order to implement the Friedman rule

By following a rule of the form

it = Φ(r_t−1− π_t)

the central bank can avoid price level indeterminacy, while setting iton average equal to zero

To see this, combine the above rule with Fisher equation it= rt+ Et{π_t+1}, which implies the difference equation

E_t{i_t+1} = Φi_t (35)

whose only stationary solution is it = 0 for all t

Under the above rule inflation is fully predictable and given by

πt = −r_t−1 (36)